Regular Polygons Quiz: Master Regular Polygons Quiz

Regular hexagons are the most efficient shape for tessellating a plane while minimizing the total perimeter for a given area. This is why bees use hexagonal cells — they require the least wax to construct while maximizing storage space. Option B is the opposite of the correct property. Options C and D are false because hexagons tessellate perfectly with no gaps.

The area of a regular polygon equals ½ times the perimeter times the apothem. This formula is derived by dividing the polygon into n congruent isosceles triangles from the center, each with base equal to one side and height equal to the apothem. Option B is the area of a triangle. Option C is the area of a circle. Option D is the area of a square only.

Perimeter = number of sides times side length = 6 times 10 = 60 cm. Option A gives 40 cm, which is the perimeter of a square with sides of 10 cm. Option B gives 50 cm, corresponding to a pentagon. Option C gives 55 cm, which has no direct relationship to a hexagon with side length 10.

The answer is True. As the number of sides of a regular polygon increases without bound, the polygon's shape approaches a circle. Each side becomes infinitesimally small and the figure becomes a smooth curve. This concept is used in calculus and geometry to derive the properties of circles from limiting cases of regular polygons.

n = 360 divided by central angle = 360 divided by 24 = 15 sides. Option A gives 12, requiring a central angle of 30°. Option C gives 18, requiring a central angle of 20°. Option D gives 24, requiring a central angle of 15°. Only 15 satisfies the given central angle of 24°.

The apothem is the perpendicular segment from the center of a regular polygon to the midpoint of one of its sides. It is used in the area formula: area = ½ times perimeter times apothem. Option A describes a radius, which connects the center to a vertex. Option B names a diagonal, which connects two non-adjacent vertices. Option D describes a line, not a specific point.

Only three regular polygons can tessellate the plane without gaps or overlaps: the equilateral triangle with interior angles of 60°, the square with 90°, and the regular hexagon with 120°. These are the only cases where the interior angle divides evenly into 360°. Option D, the regular pentagon, has interior angles of 108° which do not divide evenly into 360°, leaving gaps.

A square satisfies both conditions for a regular polygon — all four sides are equal in length and all four angles are equal at 90°. Option A only addresses angles, missing the equal sides requirement. Options C and D describe properties of squares but are not the defining conditions for regularity.

Each vertex connects to n-3 other vertices via diagonals (excluding itself and its two adjacent vertices). With n vertices, this gives n(n-3) connections, but each diagonal is counted twice so divide by 2, giving n(n-3) divided by 2. Option A gives n(n-2)/2, which overcounts by including adjacent vertices. Options C and D have no valid geometric derivation.

Central angle = 360 divided by n = 360 divided by 12 = 30°. The central angle is the angle formed at the center between two adjacent radii drawn to consecutive vertices. Option A gives 20°, corresponding to an 18-sided polygon. Option B gives 25°, corresponding to a non-integer number of sides. Option D gives 45°, corresponding to an octagon.

A regular polygon requires both equal sides and equal angles simultaneously. Option A describes only an equilateral polygon, which may have unequal angles. Option B describes only an equiangular polygon, which may have unequal sides. Option D is incorrect because right angles are specific to certain polygons like squares and rectangles, not all regular polygons.

The answer is True. For any convex polygon, regardless of the number of sides, the exterior angles always sum to exactly 360°. This can be understood by imagining walking around the polygon — a complete circuit returns you to the starting direction after turning through a total of 360°. This holds for all regular and irregular convex polygons.

n = 360 divided by exterior angle = 360 divided by 72 = 5 sides. This is a regular pentagon. Option B gives 6, requiring each exterior angle to be 60°. Option C gives 8, requiring 45°. Option D gives 10, requiring 36°. Only 5 satisfies the given exterior angle of 72°.

Each exterior angle = 360 divided by n = 360 divided by 9 = 40°. Option A gives 30°, which is the exterior angle of a regular 12-sided polygon. Option B gives 36°, which is the exterior angle of a regular decagon with n=10. Option D gives 45°, which is the exterior angle of a regular octagon with n=8.

The correct formula is (n-2) times 180. This is derived by dividing a polygon into n-2 triangles, each contributing 180°. Option A gives 180n, which overcounts. Option B gives 180(n-1), which is one triangle too many. Option D halves the result incorrectly.

An equilateral triangle has all three sides and all three angles equal, confirming A. A square has all four sides and all four angles equal at 90°, confirming B. A regular hexagon has all six sides and all six angles equal, confirming C. Option D, the rectangle, has all angles equal at 90° but sides are not all equal, so it is equiangular but not regular.

Each interior angle = (n-2) times 180 divided by n = (8-2) times 180 divided by 8 = 6 times 180 divided by 8 = 1080 divided by 8 = 135°. Option A gives 120°, which is each interior angle of a regular hexagon. Option C gives 140°, which is each interior angle of a regular nonagon. Option D gives 150°, which is each interior angle of a regular dodecagon.

Using the formula (n-2) times 180 with n=5: (5-2) times 180 = 3 times 180 = 540°. Option A gives 360°, which is the sum of exterior angles of any polygon. Option C gives 720°, which is correct for a hexagon with n=6. Option D gives 900°, which corresponds to a heptagon with n=7.

A hexagon is defined by the prefix hex meaning six. A regular hexagon has six sides all equal in length and six interior angles all equal in measure. Option A gives 4, which is a quadrilateral. Option B gives 5, which is a pentagon. Option D gives 8, which is an octagon.

The answer is False. An equilateral polygon has all sides equal but its angles may vary. A regular polygon requires both equal sides and equal angles. For example, a rhombus has all equal sides but its angles are not all equal unless it is a square, so a rhombus is equilateral but not necessarily regular.